Draw an Array Then Write a Fact Family

Fact Families (Multiplication and Division): 2.75

• Indicator of Progress
• Teaching Strategies

Supporting Materials

• Related Progression Points
• Developmental Overview of Methods of Adding (PDF - 38Kb)

Indicator of Progress

Success depends on students knowing that once they tin remember a item multiplication fact, they tin can use that fact to solve related multiplication and sectionalization tasks. The set of related facts is called a fact family.

A sample fact family unit:

seven × 5 = 35

v × vii = 35

35 ÷ 5 = seven

35 ÷ 7 = v

35 = 7 × 5

Students will already have met fact families for addition and subtraction. They will non necessarily recognise the links between multiplication and sectionalization. Students can feel overwhelmed by how many multiplication and partitioning facts there are to learn, unless they meet the links between them.

A skillful command of basic facts is required for carrying out multiplication and division algorithms.

Illustration 1: 3 different stages

Students laissez passer through three stages:

1. Students recognise the five × 7 has the same answer as 7 × 5. If they know the answer to 1, so they know the answer to the other.
2. Students can use the known fact to solve missing numbers tasks. For example, if they know 5 × 7 = 35, they can utilize this fact to solve tasks like v × ? = 35 or ? × 7 = 35.
3. Students recognise the relationship between multiplication and sectionalization. For example, if they know five × 7 = 35, then they know that 35 ÷ five = vii and 35 ÷ seven = 5.

Illustration ii

Examples of types of tasks that would be illustrative of using multiplication facts to solve multiplication problems, aligned from the Mathematics Online Interview:

• Question 28 - Sharing teddies on the mat
• Question 29 - Tennis balls chore
• Question 30 - Dot array task
• Question 31 - Teddies at the movies

Education Strategies

Understanding about fact families builds connections in mathematics and reduces the amount of material that students need to learn. The key teaching strategy is to emphasise that there are related facts that belong together. In one case a student knows one fact, they tin utilize this to solve related number sentences with missing numbers. The activities below are illustrated with the fact family of 3 × iv = 12, just teachers can employ fact families from the multiplication tables currently being learned.

Action i: Fact families from arrays uses existent objects, counters and squared paper equally representations of number facts. Students generate fact families themselves.
Activity 2: Recognising different fact families encourages students to group facts into families.
Activity 3: Fact family fortune is a simple game designed to highlight which combinations of iii numbers are in the fact family and which ones are not.
Activeness 4: Fact family bonanza encourages students to extend the notion of fact family creatively.

Activeness i: Fact families from arrays

Arrays in common objects
Many objects found at abode are bundled in arrays, for example, egg cartons, muffin trays, trays for organising nails and screws, and boxes of chocolates. Also every bit using existent objects, a digital camera tin can be used to bring pictures into the classroom.

Ask the students to choose an object or picture show and write downward as many multiplication and division number sentences as they can virtually their array. For example, consider a muffin tray which is a four by three array. Ask students to write as many number sentences as they tin can, for example, 3 × 4 = 12 and 12 ÷ 3 = 4. They should likewise write a sentence or story near these facts (e.k. there are 3 rows of muffins with 4 in each row, so in that location are 12 muffins on the tray).

Inventive students might write chemical compound sentences, although this is not the main aim of this activity (e.g. in that location are iii rows of 2 chocolate muffins and three rows of 2 plainly muffins: 3 × 2 + 3 × ii = 12).

Arrays from materials and squared newspaper
Demonstrate the fact 4 × 3 = 12 using an array of counters, centicubes, dots or symbols every bit shown below. Employ both array language (iii rows of 4 items / 4 columns of 3 items) and equal groups language (3 groups of iv items / iv groups of 3 items) so that students build links betwixt previous 'equal groups' understandings and the powerful assortment model of multiplication.

Note that the array can besides be rotated to appear as 4 rows of three, so that 4 × iii = 12 and 3 × iv = 12. Rearrangement does non change the human relationship between the 3 numbers.

Cover portions of the pictures or materials and ask questions such as "how many rows of iv makes twelve?" (? × four = 12) as a missing multiplication question and every bit a segmentation question (12 ÷ 4 = ?).

Give students some counters and get them to write other number sentences, too explained in words.

Extend to larger numbers by drawing rectangles on squared newspaper e.g. with a 1mm grid (x divisions per cm), and then by looking not at the number of small squares just at the area of the rectangle. This extends the models students accept for multiplication from equal groups, to arrays, to area of a rectangle. Link to the multiplication tables that the students are currently learning.

Action 2: Recognising dissimilar fact families

Give students a set of numbers (e.g. 3, iv, 5, 12, 15, twenty) and enquire them to write as many different multiplication or sectionalization number sentences as they can using but numbers from the set (e.1000. 3 × four = 12 and 12 ÷ 4 = 3). Inquire students to group all the number sentences from the same family together. For case, here are viii number sentences from one fact family (there are two more fact families possible with the numbers provided):

iii × 4 = 12	4 × three = 12	12 ÷ four = 3	12 ÷ 3 = four
12 = 3 × 4	12 = 4 × 3	3 = 12 ÷ iv	iii = 12 ÷ four

Note that the last row contains number sentences that some students think are 'backwards'. They are just equally valid every bit the number sentences in the top row, and it is important that these are included to help students with the notion that an equals sign can mean 'balance' every bit well as 'give an answer'. For more than information encounter: The meaning of the equals sign. (Level 2.25) which addresses the   demand for students recognising that the expressions on either side of an equation have the same value.

Include multiplication facts that the students are currently learning. Stress the usefulness of these links in calculating: "I tin can't remember 9 sixes, but I exercise know 6 nines."

Activity three: Fact family fortune

Here is a listing of all the missing number sentences associated with the three by 4 assortment. The first column contains the fact family unit, while the number sentences in the other columns have a missing number. Students need to experience all of these different combinations at some stage.

four × 3 = 12	4 × iii = □	4 × □ = 12	□ × 3 = 12
iii × 4 = 12	3 × iv = □	iii × □ = 12	□ × four = 12
12 ÷ iv = 3	12 ÷ 4 = □	12 ÷ □ = 3	□ ÷ iv = iii
12 ÷ 3 = 4	12 ÷ 3 = □	12 ÷ □ = 4	□ ÷ 3 = 4
12 = 4 × 3	12 = 4 × □	12 = □ × 3	□ = 4 × 3
12 = iii × 4	12 = iii × □	12 = □ × iv	□ = three × iv
three = 12 ÷ iv	3 = 12 ÷ □	three = □ ÷ 4	□ = 12 ÷ iv
four = 12 ÷ 3	4 = 12 ÷ □	4 = □ ÷ 3	□ = 12 ÷ 3

Be sure that students sympathise why some combinations are Non in the fact family. For example:

3 ÷ 12 = 4

4 ÷ 12 = 3

4 ÷ 3 = □

□ × 3 = 4

Game: Fact family unit fortune
Students make themselves two ready of three cards, and a resource sail with 2 columns: Truthful/ FALSE.

One set up (for example, coloured red) contains:

• a bill of fare with 12 written on ane side
• a card with iii written on i side and
• a card with four written on one side.

The other gear up of three cards (for example, coloured blue) contains:

• a card with = on one side
• a menu with × on one side and
• a card with ÷ on one side.

• Shuffle each gear up, face down.
• Deal a red card, a blue carte du jour, a red carte du jour, a blue card and a reddish card along the table.
• Turn over and read the number sentence, for example 3 ÷ 12 = 4. (If there is no = in the number sentence, bargain over again).
• Write this 'fact' down under either the heading Truthful or the heading FALSE. In this case, it goes under FALSE.
• Students talk over why this is the case.
• Pick upwards all the cards, shuffle, and play again.
• Place the resulting fact under the appropriate heading e.1000. 3 = 12 ÷ iv goes nether the heading Truthful.
• Continue making the list.

Finish the activeness with a discussion of which facts are true and which are faux.

Activity 4: Fact family bonanza

There is no need to constrain students' understanding of fact families to a express set. The main thought is that many number facts are related, and this makes adding easier.

This is an open activity where students can exercise creativity and work at their own levels.

Brainstorm with a multiplication argument at any appropriate level (east.g. five × 7 = 35, eleven × 8 = 88, 77 × 13 = 1001).

Students make as many facts equally they can from this one fact, saying how it is derived from the basic fact. As well as standard 'fact family' variations, students may include 11 × 80 = 880, 11 × 16 = twice 88 = 176, 110 x 8 = 880 and division variants, as well equally derived facts such every bit 12 × 8 = 88 + eight = 96 etc).

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Source: http://smartvic.com/teacher/mdc/number/N27501P.html

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