# Equal sets (easy multiplication and division)

Purpose

In this unit students explore multiplication to find the total of ‘sets of equal sets.’ Students also encounter problems with division through forming equal sets.

Achievement Objectives
NA2-1: Use simple additive strategies with whole numbers and fractions.
NA2-2: Know forward and backward counting sequences with whole numbers to at least 1000.
Specific Learning Outcomes
• Use multiplication by two, five and ten to solve problems with equal sets.
• Use multiplication (as above) to solve division problems with equal sets.
• Pose multiplication and division problems.
• Record working with multiplication and division using diagrams and equations.
Description of Mathematics

The basic concept of multiplication is an important one because of its practicality (How much do 4 ice creams cost at \$2 each) and efficiency (It is quicker to determine 4 x 2 than to calculate 2 + 2 + 2 + 2). Multiplication is used in many different situations. Here students think about multiplication as a short way to find the result of repeated addition of equal sets.

A rate problem involves a statement of ‘so many of one quantity for so many of another quantity.’ All multiplication situations contain some form of rate but at this level, the problems are usually about equal sets or measurement. Take this example:

“Lena buys six bags of biscuits. Each bag contains four biscuits. How many biscuits does she buy altogether?”

This is an equal sets problem that contains the rate ‘four biscuits for every bag.’ A measurement rate problem is usually something like this:

“Hone’s kumara plant grows five centimetres each week after it sprouts. How long will his plant be after six weeks?”

The rate in Hone’s problem is “five centimetres for every week.”

As well as thinking about multiplication in a variety of situations, students are encouraged to use a variety of materials and mathematical equations to solve the problems. These representations help students see the multiplicative structure that is common to a variety of problems and assist them to transfer their understanding to new-to-them situations. Transfer to new situations is also made easier for students by learning to record their working and answers using diagrams and equations. For example, if two different situations are modelled by 4 x 3 = 12 students might investigate what is the same and what is different about those two situations.

Division is the inverse operation to multiplication which means division undoes multiplication and vice versa. Start with four for example. 3 x 4 = 12, then 12 ÷ 3 = 4. The effect of multiplying by three then dividing by three is to leave four unchanged.

Division has two forms partitive (sharing) and quotative (measuring). Students tend to be more familiar with sharing situations, like, 12 things are shared equally into three sets. How many things are in each set?

This unit develops measurement division, using multiplication facts. A typical measurement division problem is something like, 12 things are put into sets of three. How many equal sets are made?

This unit can be differentiated by varying the scaffolding provided to make the learning opportunities accessible to a range of learners. For example:

• Accept students’ use of counting and equal addition strategies to solve multiplicative problems, while encouraging them to learn and use multiplication facts.
• Have students use materials or diagrams to support their thinking.
• Use strategies like masking materials, and ‘think what will happen’, to encourage students to anticipate the results of actions.
• Work in small groups with students who need additional support, solving problems together.
• Use collaborative groups with students of mixed, but not too divergent, achievement so students can support each other.

This unit uses an authentic science context of kōura (freshwater crayfish). If the context is inappropriate for your students, choose familiar contexts that include multiplicative situations to appeal to students’ interests and experiences and encourage engagement. Examples may include:

• lines of students in kapa haka groups
• collecting bags of pipi or other shellfish
• crews of students racing in waka ama
• loaves of bread for a school or community event
• groups of people travelling in vans, cars or buses
• preparing bundles of harakeke for weaving.
Required Resource Materials
Activity

#### Session One

1. Use PowerPoint One to introduce the scenario about kōura (freshwater crayfish). Videos of kōura can be found online to motivate and inform students. You may wish to look at the Department of Conservation page on kōura.
2. The problems are based around an equal number of kōura in each pond. Use think (individual), pair (with a partner), share (whole class) to work on each problem. Allow students access to materials such as cubes, counters, toy creatures, etc.
3. In whole class discussions look to highlight these points:
• Equal addition can be used but multiplication is more efficient, for example, knowing 6 tens equal 60 is better than counting 10, 20, 30, …, 60.
• Equal groups can be grouped to make calculation easier, for example, 8 fives can be regrouped as 4 tens.
• Addition and multiplication equations can be used to represent the same situation, for example, 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 = 18 can be written as 9 x 2 = 18. Note that the first factor is the multiplier, representing the number of equal sets.
4. Encourage students to share their calculation strategies and record their calculations using addition and multiplication equations. Emphasise the brevity of multiplication equations compared to the equivalent repeated addition equation.
5. Slide Eight has a night-time scenario to encourage imaging. Watch to see how students individually create a problem.
• Do they create easier factors like two, five, or ten?
• Are they clear of the meaning of each factor? (Multiplier is the number of ponds and the multiplicand is the number of kōura in each pond).
6. Ask students to share their problem with a partner. Choose a couple of problems for the whole class to solve. Slide Nine can be used to create pictures of problem situations.
7. Provide students with Copymaster One to work on in pairs. After a suitable time gather the class to share their answers. Highlight the following:
• Use the meaning of factors to compare total numbers, for example, 4 x 9 is greater than 9 x 3.
• Use a systematic approach to find missing factors, for example, 1 x 20, 2 x 10, 4 x 5. Will [ ] ×3=20 have any whole number answers?

#### Session Two

In this session students build up their knowledge of x2, x5, and x10 facts and the commutative property. Remembering those basic facts gives them knowledge from which they can create ‘new’ facts.

1. Introduce subitising cards for two, five, and ten kōura (Copymaster Two). Flash the cards at your students to see if they can instantly say the number of kōura, hold up that many fingers, write the number in the air, etc.
2. Put your students into groups of three to quickly cut up their set of cards (laminating is ideal but not necessary). Ten of each card is sufficient for a trio. Ask the students to put the cards in stacks of the same number, twos, fives, and tens.
“We’re going to play a game called Make That Many. I will write a number and I want your group to make that many kōura with your cards.”
3. Write up these numbers, one at a time, with students making the number with cards.
10               12               20                   15               13                   25
4. After each number record the ways students create using equations. Note that many ways do not use multiplication, but many do. Examples are below:
10 – 1 x 10, 5 + 5 (2 x 5), 2 + 2 + 2 + 2 + 2 (5 x 2)
12 – 10 + 2, 5 + 5 + 2, 2 + 2 + 2 + 2 + 2 + 2 (6 x 2)
20 – 10 + 10 (2 x 10), 10 + 5 + 5, 5 + 5 + 5 + 5 (4 x 5),
2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 (10 x 2), etc.
15 – 10 + 5, 5 + 5 + 5 (3 x 5), 2 + 2 + 2 + 2 + 2 + 5
13 – 2 + 2 + 2 + 2 + 5, A prime number, not possible with x2, x5 or x 10.
25 – 10 + 10 + 5, 5 + 5 + 5 + 5 + 5 (5 x 5), etc.
5. Write only the multiplication solutions on separate cards:
1 x 10 = 10, 2 x 5 = 10, 5 x 2 = 10, 6 x 2 = 12, 2 x 10 = 20,
4 x 5 = 20, 10 x 2 = 20, 3 x 5 = 15, 5 x 5 = 25
6. Put related cards together, such as;
3 x 5 = 15 and 4 x 5 = 20
2 x 10 = 20 and 4 x 5 = 20
2 x 5 = 10 and 4 x 5 = 20
5 x 2 = 10 and 6 x 2= 12
7. Discuss:
What patterns do you notice?
Can you explain why the patterns happen?
Look for students to notice similarity and difference.
For example, 5 x 2 = 10 and 6 x 2 = 12. “Both equations have equal sets of two but 6 x 2 has one more set of two than 5 x 2.”
8. Physically model the equations with kōura cards to support students to connect equations with their meanings.
9. Show your students how to play the Kōura game in threes, using their cards.
• One player makes a multiplication fact using kōura and ponds. The ponds can be containers, or pieces of paper. The equal set cards are hidden under the ponds. For example, Player 1 makes four ponds with a ten kōura card in each pond.
• The player reveals the contents of only one pond then re-hides the kōura card.
• The other two players work out the total number of kōura. Each player gets a point if they are correct.
• Players take turns making and guessing.
• The player with the most points wins.
10. Record the tens facts in a list and ask students to recite the list in order. Gradually erase facts but expect students to recite the full list. Encourage students to notice that the “ten times” tables are just the ‘ty’ numbers. For example, ten times six equals sixty.
Similarly, eliminate facts in the “five times” and “two times” tables. You might continue this over many days.
11. Copymaster Three is a game that supports learning the x2, x5, and x10 facts. Let students play the game in threes. They will need:
• A die with faces labelled “2, 2, 5, 5, 10, Choose”, or a set of cards with the same labels (four of each).
• Counters to cover the numbers.
Players take turns to:
1. Roll the dice.
2. Name a multiplication using the factor they roll that will match an uncovered number, For example, 8 x 2 = 16.
3. Cover the number on their board that matches the product (answer).
4. If ‘Choose’ comes up they can nominate 2, 5, or 10 as their chosen factor.

If the multiplication fact is incorrect, the counter is removed, and the player must wait for their next turn.
The first player to cover all their numbers is the winner.

12. Another useful practice task involves the calculator. Make sure that the calculator is a simple four function model that is commonly cheap to buy.
These calculators have a constant function. With multiplication the calculator remembers the first factor as long as nothing is cleared.
Key in 2 x 6 = to get 12.
Enter a new number such as 10 (no clearing) and press = . You get 20, that is, 2 x 10 (the calculator remembers 2 x ).
Try entering other numbers and pressing =. The result is always 2 x the number you put in until you use a C or AC button.
13. Students play “How many times?” with a partner (One calculator between two students)
Take turns to:
• Player A enters a 2x , 5x,or 10x fact, e.g. 5 x 8 = 40.
They hand the calculator to Player B.
• Player B see only the product (answer) to start.
They try different factors until they get the target number.
For example, they try 6 = and get 30. Then they try 8 and get the target 40.
• Players keep a tally of how many tries it takes to reach the target. If you want a competitive game, the player with the least number of tries, after many rounds, wins,

#### Session Three

In this session students explore how known x2, x5, and x10 facts can be used to work out other multiplication problems.

1. Use PowerPoint Two in the start of the lesson. Slides 1-6 pose can be used to illustrate how to derive new facts from x2 and x5 facts.
For example, Slide One shows five ponds with two kōura in each pond. Clicking on the animation will add one kōura to each pond.
How many kōura are in the five ponds altogether?
How do we write that as an equation? (5 x 2 = 10)
If we add one more kōura to each pond, what will we have?
(Five ponds with three kōura in each, 15 kōura in total, 5 x 3 =15)
2. Other examples are:
Slide Two: 8 x 2 = 16 so 8 x 3 = 24
Slide Three: 7 x 2 = 14 so 7 x 3 = 21 (Masked until animated with a mouse click)
Slide Four: 4 x 5 = 20 so 4 x 6 = 24
Slide Five: 8 x 5 = 40 so 8 x 6 = 48 (Masked until animated with a mouse click)
3. Give your students Copymaster Four to work on in pairs. Roam the room to look for:
Do your students recall their basic x2, x5, and x10 facts or do they still calculate them using skip counting or repeated addition?
Do they use a known fact to work out a new fact?

#### Session Four

In this session students use multiplication to anticipate the result of repeated subtraction, known as quotative division. For example, one interpretation of 12 ÷ 3 is “How many threes can be subtracted from 12?” Students continue to use the kōura and ponds scenario.

1. Use PowerPoint Three to lead the discussion. You might show part of the Country Calendar episode (TVNZ OnDemand login required) devoted to farming koura to tell John’s story. Students might use the cards from Session Three to work out the problems if needed. Anticipating the answer is preferable.
Key points to bring out are:
• Allocating equal numbers of koura to ponds is repeated subtraction, for example, 20 – 5 – 5 …
• Anticipating the number of equals sets that can be subtracted can be solved inefficiently by counting backwards or subtracting, for example, 20 – 5 = 15, 15 – 5 = 10, etc.
• Using multiplication is more efficient, for example, 4 x 5 = 20 anticipates the number of fives that can be subtracted from 20.
• Repeated subtraction can be written as division, for example, 20 ÷ 5 = 4, read as “Twenty divided by five equals four.”
2. Slide Nine shows pairs of equation, first multiplication then second, the corresponding division. Note that the equations represent a repeated subtraction of equal sets view of division.
What was the problem that went with these equations?
Focus on the meaning of the numbers and operation symbols in each equation.
What does the ÷ sign mean in these problems?
3. In general, agree that ÷ means “measured in equal sets of” to get the idea that the number of times one set (the divisor) fits into another (the dividend or first number).
4. Use Copymaster Five to make the game Fill the Ponds. The rules are as follows:
You need: 1 pond board each, a set of cards between two players, coloured counters to cover the numbers.
This is a game for two players, but it can be played in fours in two sets of cards are used.
To start, shuffle the cards and deal each player three cards. Take turns to:
• Choose one of your cards, e.g. 20.
• Choose the divisor (kōura per pond) from the numbers 2, 5 and 10, e.g. Choose 2.
• If you put [card number] koura into ponds, [chose divisor] in each pond, how many ponds do you fill? e.g. 20 ÷ 2 = 10.
• Cover the answer on you board, e.g. Cover 10 on your board.
• Take another card from the deck so you start the next turn with three cards. Put the card you used to the bottom of the deck.
5. The first player to cover all their numbers wins.
6. Discuss the game.
Why are there more of some numbers than others on the board?
Which numbers were hard to get? Why?
Which calculations were easy? Which calculations were hard?
What makes the calculations easy or hard?
7. Let students practise division using Copymaster Six. They can work individually or in pairs. Look particularly at the way students solve the division problems:
Do they use multiplication facts, or rely on counting or repeated addition/subtraction?

#### Session Five

Devote this session to three activities:

1. Students write 'kōura in ponds' problems for others to solve.
2. Students complete unfinished work, replay games, and learn basic facts (x2, x5, and x10).
3. Use the time when students work independently to assess achievement.

You might use the following problems as examples to use in mini-interviews. Let students use pencil and paper to solve the problems. As well as discussion, their jottings will give useful insight into their preferred strategies.

The focus of observation for each question is also included below.

1. There are eight ponds. Five kōura live in each pond.
How many kōura is that altogether?
This problem is 8 x 5 = [     ] (product unknown).
Do students use their multiplication facts or fall back to counting or repeated addition?
2. John has 60 kōura to seed some ponds.
He wants ten kōura in each pond.
How many ponds does he need to find?
This problem is [     ] x 10 = 60 (multiplier unknown). It can also be represented as division, i.e. 60 ÷10 = [    ].
Do students use their knowledge of x 10 facts to solve it?
3. There are nine ponds and18 kōura altogether.
The same number of kōura live in each pond.
How many kōura are in each pond?
This problem can be represented as 9 x [    ] = 18 or 18 ÷ 9 = [    ]. Note that this is sharing (partitive) division.
4. Imagine seven ponds, each pond has five kōura.
How many kōura is that altogether?
Imagine you put two more kōura in each pond.
How many kōura would you have then?
This problem assesses students’ ability to derive facts. It can be written mathematically as 7 x 5 = 35 so 7 x 7 = 49.
5. If 8 x 10 = 80, what does 8 x 11 equal?
Similarly this item checks to see if students recognise the meaning of multiplication equations, that is, 8 x 10 as eight sets of ten, and their ability to derive new facts, that is, eight set of eleven results in eight more.
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