Multiplication, Division and Equal Groups

Ready-to-teach lesson

New Mexico edition

TUMBLETA MATH UNIT | NEW MEXICO

Multiplication, Division and Equal Groups

Grade 3 | 10 lessons | 30 minutes each | 4 books

ABOUT THIS UNIT

This unit brings together carefully selected lessons from several TumbleMath books to build one connected mathematical idea. Each book becomes a starting point for modelling, discussion, and visible student work. Teach the sequence as a complete unit, or choose individual lessons to support your existing program. Assessment checkpoints, a culminating task, and a reproducible student page are included.

THE MATHEMATICAL ARC

Students move from skip-counted equal groups to multiplication equations, arrays, and fair division.

The final lessons vary which quantity is unknown so students must interpret the situation rather than hunt for a keyword.

Source books

• Count on Pablo

• 2 X 2 = Boo!

• Dinosaur Deals

• The Multiplying Menace Divides

HOW TO USE THIS GUIDE

Each 20-minute daily core comes from the named source lesson. This guide supplies the sequence, five-minute connections, five-minute closes, checkpoints, and culminating task. Open the complete source lesson before teaching and use its listed materials and student product.

Where This Fits in Your Year

Recommended

Grade 3, during initial multiplication and division work and before

placement

complete fact fluency.

Students arrive

Students can form equal groups, skip-count, and connect repeated

knowing

addition to a total.

page 1

This unit develops

Students connect equal groups to multiplication equations and arrays, then interpret division by identifying which grouping quantity is unknown.

This prepares

Fluency within 100, area reasoning, and multi-step multiplicative

students for

problems.

PLACEMENT NOTE

Curriculum maps differ by district, board, division, and school. Place the unit where students have the listed prerequisites and can use the culminating task as evidence of the stated expectations.

STRONGEST CURRICULUM CONNECTIONS - OUR SUMMARY

New Mexico Mathematics Common Core Standards | Grade 3

Direct unit-level expectations

3.OA.A.1

Interpret products of whole numbers, e.g., interpret 5 × 7 as the total number of objects in 5 groups of 7 objects each.

Unit evidence: Lessons 1-7 interpret products through equal groups and arrays.

3.OA.A.2

Interpret whole-number quotients of whole numbers, e.g., interpret 56 ÷ 8 as the number of objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each.

Unit evidence: Lessons 8-10 distinguish both meanings of a quotient.

3.OA.B.6

Understand division as an unknown-factor problem.

Unit evidence: The final lessons derive and check quotients with related multiplication.

Additional expectation partially supported

3.OA.A.3 | Partial support

Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.

The unit directly teaches equal-group and array situations, but it does not teach the measurement-quantity problem type included in this expectation.

page 2

Unit at a glance

Teach the lessons in order. Use the complete source lesson for the 20-minute core. The five-minute opening and five-minute close carry the sequence from one day to the next. Each lesson also includes a standalone opening for teachers who use one lesson outside the sequence.

page 3

Unit vocabulary

equal group, groups of, factor, product, array, row, column, divide, quotient, total, number of groups, amount in each group

Materials

Complete source lessons and their named student pages; counters; connecting cubes; small bags or cups; grid paper; row mats; equation cards; the mathematical evidence sheet reproduced in this guide.

Assessment plan

The assessment loop is intentionally light: review student work every day, check the meaning of equal groups after Lesson 3, and check the first division interpretation after Lesson 8. Finish with one performance task.

Daily evidence to preserve

• A visible model or representation, not an answer alone.

• An equation or comparison that agrees with the model.

page 4

• Labels or a sentence identifying what each number counts.

• A correction when the student changes a claim after testing it.

INCLUSION MOVE

Let students explain with objects, drawings, gestures, oral language, or writing. Keep the mathematics constant: the quantities, relationships, and evidence must remain visible.

LESSON 1 · EQUAL GROUPS AND SKIP COUNTING

Count the Onions by Twos

Source book: Count on Pablo | 30 minutes

I CAN

I can count equal groups of two.

BIG IDEA

Skip-counting works because each group has the same size.

SUCCESS

The grouping, count sequence, and total agree.

• 1.Connect | 5 minutes Put out a loose pile of counters and ask how many. Then ask for a faster

way than one at a time. Take the suggestions and write them where everyone can see them.

Today tests which of them can be trusted.

Taught on its own: Put out a loose pile of counters and ask students to organize it into equal groups of two, then count the groups to find the total.

• 2.Source lesson | 20 minutes Organize objects in pairs and skip-count to find the total. Teach the

full source lesson as written, including its named materials and student product.

• 3.Close | 5 minutes Students finish: I made ___ groups of two and counted ___, so the total is

___. Ask what would go wrong in the count if one group held three.

WATCH FOR

A count by twos continuing across a group that only holds one.

LESSON 2 · GROUPS OF FIVE

Five in Each Bag

Source book: Count on Pablo | 30 minutes

I CAN

I can find a total from equal groups of five.

page 5

BIG IDEA

Each group must contain the same number.

SUCCESS

The student identifies group size, number of groups, and total.

• 1.Connect | 5 minutes Yesterday every group held two and the count ran 2, 4, 6. Ask what the

count would have been if one group had held three instead.

Today the groups hold five, and the same rule about sameness decides whether the count works.

Taught on its own: Put out three bags holding five, four, and five counters and ask which bag breaks the count.

• 2.Source lesson | 20 minutes Build equal bags of five and connect the groups to a repeated

count. Teach the full source lesson as written, including its named materials and student product.

• 3.Close | 5 minutes Students finish: I had ___ groups of five, so the total is ___. The number of

groups and the number in each group have to be said separately, not folded into one number.

WATCH FOR

A total reported without saying how many groups produced it.

LESSON 3 · INTERPRET PRODUCTS

Three Scarecrows, Same Parts Each

Source book: 2 X 2 = Boo! | 30 minutes

I CAN

I can write an equation for equal groups.

BIG IDEA

A product describes equal groups: number of groups times amount in each.

SUCCESS

The model and factors match the stated group meanings.

• 1.Connect | 5 minutes Yesterday's sentence named two different things: how many groups, and

how many in each. Ask a student to say both halves of theirs again. Today those two numbers get written as a multiplication equation, and each one keeps its job.

Taught on its own: Build three equal groups of four where everyone can see them and ask for two numbers that describe the arrangement.

• 2.Source lesson | 20 minutes Build three equal groups and write a multiplication equation for

repeated features. Teach the full source lesson as written, including its named materials and student product.

page 6

• 3.Close | 5 minutes Students finish: ___ groups of ___ is ___ × ___ = ___. Ask which factor

counts the groups and which counts what is inside them.

WATCH FOR

Factors written in an order the student cannot explain.

CHECKPOINT A

Give one new, book-free problem with the same mathematical structure. Score only: representation, accurate relationship, and explanation. Reteach with objects if a student can calculate but cannot label the groups. Use the result to group tomorrow's opening.

LESSON 4 · DOUBLING AS MULTIPLICATION

Two Rows in the Spectacles

Source book: 2 X 2 = Boo! | 30 minutes

I CAN

I can connect doubling, an array, and a multiplication equation.

BIG IDEA

Two equal rows can be described as a double and as multiplication.

SUCCESS

Both rows are equal and the equation matches the array.

• 1.Connect | 5 minutes Yesterday each factor had a job. Ask a student to point at the

group-counting factor in their equation. Today there are exactly two groups, and they get arranged as rows so the doubling can be seen rather than asserted.

Taught on its own: Lay out two equal rows of counters and ask for two different ways to name the total.

• 2.Source lesson | 20 minutes Arrange objects in two equal rows and connect doubling to

multiplication by two. Teach the full source lesson as written, including its named materials and student product.

• 3.Close | 5 minutes Students finish: Two rows of ___ is ___, and doubling ___ gives the same

___. The rows and the doubling both have to appear in the writing.

WATCH FOR

Rows drawn at different lengths and still called a double.

page 7

LESSON 5 · ARRAYS FOR MULTIPLICATION

Rows of Four: Drawing What a Trade Costs

Source book: Dinosaur Deals | 30 minutes

I CAN

I can draw an array for a multiplication situation.

BIG IDEA

Rows make both the number of groups and the amount in each group visible.

SUCCESS

Rows, items per row, total, and equation all agree.

• 1.Connect | 5 minutes Yesterday two equal rows showed a double. Ask what would change if a

third row were added, and what would stay the same. Today the rows keep going, and the drawing has to make both factors readable without counting every item.

Taught on its own: Show a loose pile of twelve counters and ask students to arrange it so the total can be seen without counting one at a time.

• 2.Source lesson | 20 minutes Represent repeated trade costs as equal rows of four. Teach the

full source lesson as written, including its named materials and student product.

• 3.Close | 5 minutes Students finish: My array has ___ rows of ___, so ___ × ___ = ___. The rows

have to be equal on the page, not only in the sentence.

WATCH FOR

A ragged array whose second factor cannot be read off the drawing.

LESSON 6 · FLUENCY FROM GROUPS

Count the Price: Fours and Threes at Speed

Source book: Dinosaur Deals | 30 minutes

I CAN

I can find products of three and four efficiently.

BIG IDEA

Fluent facts can still be explained with groups.

SUCCESS

An accurate fact plus a group or array explanation.

• 1.Connect | 5 minutes Yesterday an array made both factors readable at a glance. Ask a student

how many items their array held and how they knew without counting all of them. Today the facts come faster, and the meaning still has to be available on request.

page 8

Taught on its own: Ask for 4 × 3, then ask what the four counts and what the three counts.

• 2.Source lesson | 20 minutes Use equal-group structures to skip-count by threes and fours while

preserving meaning. Teach the full source lesson as written, including its named materials and student product.

• 3.Close | 5 minutes Students finish: ___ × ___ = ___, and I know it because ___. A student who

answers quickly must still be able to name the groups behind the answer.

WATCH FOR

A fast correct answer with no group or array behind it when asked.

LESSON 7 · MULTIPLICATION BY FIVE

Boo Stew for Five

Source book: 2 X 2 = Boo! | 30 minutes

I CAN

I can use equal groups to multiply by five.

BIG IDEA

Multiplying by five counts equal groups of five.

SUCCESS

The sequence 1 × 5 through 4 × 5 matches the models.

• 1.Connect | 5 minutes Yesterday speed was allowed as long as the meaning could be produced

on request. Ask one student to produce the meaning behind a fact they said quickly. On lesson 2 the groups of five were already made and waiting to be counted; today they are built one at a time, so the growth between products is visible.

Taught on its own: Build one group of five, then a second, and ask what changes and by how much each time.

• 2.Source lesson | 20 minutes Build one through four equal groups of five and record each

product. Teach the full source lesson as written, including its named materials and student product.

• 3.Close | 5 minutes Students finish: ___ groups of five is ___, which is five more than ___ groups

of five. Each new group has to add five, and the sentence has to show where.

WATCH FOR

Products of five recited as a chant with no groups underneath them.

page 9

LESSON 8 · INTERPRET DIVISION

Three Sets of Four: What the Spell Does to

Twelve

Source book: The Multiplying Menace Divides | 30 minutes

I CAN

I can interpret a quotient using equal groups.

BIG IDEA

Division can ask for the number of groups or the amount in each group.

SUCCESS

The student labels total, number of groups, and amount per group.

• 1.Connect | 5 minutes For seven days the groups were known and the total was the thing worked

out. Ask what has been given every time so far, and what has been found. Today the total is given first and the groups are what is missing.

Taught on its own: Put out twelve counters and ask students to share them into three equal groups, then say what the three counts and what the four counts.

• 2.Source lesson | 20 minutes Partition 12 as three groups of four and four groups of three,

naming what each quotient means. Teach the full source lesson as written, including its named materials and student product.

• 3.Close | 5 minutes Students finish: ___ shared into ___ groups gives ___ in each group. Ask

what the quotient counts in their sentence: the groups, or what sits inside one.

WATCH FOR

A quotient read as the number of groups when the problem asked for the amount in each, or the reverse.

CHECKPOINT B

Give one situation and ask for it twice: once as equal groups and once as an equation. Ask the student to name what the quotient counts. Score only: representation, accurate relationship, and explanation. Reteach the relationship, not a memorized procedure, when the model and the equation disagree.

page 10

LESSON 9 · ARRAYS CONNECT OPERATIONS

Five Rows of Three: Drawing Peter's Fifteen

Holes

Source book: The Multiplying Menace Divides | 30 minutes

I CAN

I can write related equations from an array.

BIG IDEA

One array can represent related multiplication and division facts.

SUCCESS

A correct array and at least two related equations.

• 1.Connect | 5 minutes Yesterday a total was split into equal groups. Ask whether yesterday's

three numbers could also have been written as a multiplication.

Today one array is asked to produce both kinds of equation without being redrawn.

Taught on its own: Draw an array of five rows of three and ask what division question it could answer.

• 2.Source lesson | 20 minutes Build an array for a division situation and write the related

multiplication and division equations. Teach the full source lesson as written, including its named materials and student product.

• 3.Close | 5 minutes Students finish: My array shows ___ × ___ = ___ and ___ ÷ ___ = ___. Both

equations have to come off the same drawing: the rows give one, and reading down a column gives the other without a new count.

WATCH FOR

A division equation worked out separately instead of read from the array already drawn.

LESSON 10 · UNKNOWN NUMBER OF GROUPS

How Many Groups Made Four Frogs?

Source book: The Multiplying Menace Divides | 30 minutes

I CAN

I can find a missing number of groups and check it.

BIG IDEA

The total and the amount in each group determine how many groups there are.

page 11

SUCCESS

A correct number of groups, a matching model, and a multiplication check.

• 1.Connect | 5 minutes Yesterday one array carried a multiplication and a division at the same

time. Ask which number in that array the division was looking for.

Today that number is missing from the start, and the array has to be built to find it.

Taught on its own: Show twenty counters and bags that hold four each, and ask how many bags will be filled.

• 2.Source lesson | 20 minutes Use the total and the amount in each group to determine how many

groups there are. Teach the full source lesson as written, including its named materials and student product.

• 3.Close | 5 minutes Students finish: I knew ___ in all and ___ in each group, so there were ___

groups, and I checked by ___. The check has to be a multiplication, not a second count.

WATCH FOR

The answer recounted as the check instead of multiplied back.

Culminating performance task

A creature keeper has 24 food pieces. Students build or draw two genuinely different multiplication arrays for 24: 3 × 8 and 4 × 6. They write the related multiplication and division equations, then solve:

(a) 4 pieces per bowl - how many bowls? (b) 6 bowls - how many pieces per bowl? (c) 3 pieces per bowl - how many bowls? Finally, they explain how the two arrays support the three division answers.

Student directions

• Build or draw both arrays so another student can check them.

• Write the matching multiplication and division equations.

• Label what every number counts.

• Explain one relationship using the model as evidence.

page 12

TEACHER KEY

The required arrays are 3 × 8 and 4 × 6. Turning an array does not create a second array. Related equations include 3 × 8 = 24, 8 × 3 = 24, 24 ÷ 3 = 8, 24 ÷ 8 = 3, 4 × 6 = 24, 6 × 4 = 24, 24 ÷ 4 = 6, and 24 ÷ 6 = 4.

In part (a), 6 counts bowls. In part (b), 4 counts pieces in each bowl. In part (c), 8 counts bowls, so the 3 × 8 array supplies evidence for a division question. Both required arrays stay within the Grade 3 multiplication range used in this unit.

TumbleTA Math Unit · Launch Collection

Reproducible | Mathematical evidence sheet

Name ____________________________________________________ Date ____________________

Problem or claim

My model

page 13

My equation or comparison

My explanation

Teacher planning and handoff

Before teaching

• Open each named source lesson and print only its listed student pages.

• Prepare the concrete materials named in the source lesson; do not substitute a picture when the

proof depends on moving or grouping objects.

page 14

• Choose whether the final explanation will be oral, written, or recorded.

If time is tight

• Keep the 20-minute core intact. Shorten the opening to its first question and collect the source

product as the close.

• Do not remove the model-building step; it is the evidence base for the unit.

If you teach one lesson on its own

Use the standalone opening printed under Connect instead of the sequenced opening. Everything else in the lesson stands alone.

Using the New Mexico expectations

Use 3.OA.A.1, 3.OA.A.2, 3.OA.B.6 as the direct unit-level curriculum connections. 3.OA.A.3 is included only as partial support; its omitted requirement must be taught elsewhere.

TEACHER NOTE

You know your students and what they need today. Use what works, change what does not, and preserve the mathematical evidence: the quantities, equal-group relationships, models, and explanations.

page 15