Multiplication

This is a four week unit on Multiplication and Division. We will be spending 2 weeks on Multiplication and 2 weeks on Division. These are the Core Compnents that must be taught:

· Introduces multiplication and division in problem situations that can be directly modeled. · Models multiplication as repeated addition, rectangular arrays, and equal groups. · Models by skip counting on a number line. · Builds set models. (Ex. ☺☺☺ ☺☺☺ is 2 groups of 3.) · Builds rectangular arrays (area models). · Uses invented methods or standard algorithms such as partial products, or the lattice method to solve two digit multiplication problems. · Relates drawings and models to expressions. (Ex. 2 x 5 is 2 groups of 5) · Uses known facts to derive unknown facts. (Ex. If 5 x 3 = 15, then 6 x 3 = 15 + 3 more; If 3 x 3 = 9, then 6 x 3 is 9 + 9.) · Completes multiplication chart. · Understands that operations for multiplication and division are inverse operations ( 3X2 = 6 and 6÷2 = 3) · Understands and uses the commutative property of multiplication to find another member of the fact family (If 3 X 2 = 6 then 2 X 3 = 6) · Understands that division represents sharing equally or forming equal groups. · Draws pictures to solve problems. · Writes an equation (number sentence) to represent a multiplication or division situation. · Finds the missing numbers in a number sentence. · Understands that not all numbers can be divided equally resulting in a remainder. · Estimates solution before solving · Uses standard form and expanded form (breaks apart numbers before multiplying). · Uses computers, calculators, and other technology to explore multiplication patterns. · Works with tables of related number pairs that may not begin with one and/or may not be sequential. · Displays data in both horizontal and vertical tables. · Identifies horizontal patterns in a table, as well as vertical. Describes the patterns verbally and in writing.

Introducing multiplication: Use the following book, "What Comes in 2's, 3's, and 4's?" by Suzanne Aker.

**Day One:**

Having children investigate groups of objects in a real-world context helps them link the idea of multiplication to the world around them. It also helps them avoid seeing mathematics as totally abstract and unrelated to their lives.

**The chopstick problem:** Use this problem to initiate the exploration of multiplication in a context. First, make sure children know that when people use chopsticks to eat, two are required. Then pose a problem for class discussion: //How many chopsticks are needed for four people?// Hear from all children who want to respond, asking them to explai how they arrived at their answers.

Then pose another problem: //How many chopsticks are needed for everyone in the class?// Ask children to discuss and solve this problem in small groups. Then have individuals report their answers, again asking them to explain their reasoning. Record on the board the methods they report, modeling for the children how to use mathematical notation to represent their ideas.

1. Read the book to the class.

2. Have the students brainstorm other things that can be grouped in pairs. Make a chart on butcher paper and begin to list students' ideas. For example, eyes, hands, ears, shoes, arms, dancing couples, etc. Students can record the class ideas on their own chart to put into their Math Journals.



3. Have students think of things that are grouped in 3's. Record two or three responses and then put students into groups. Direct each group to continue thinking of more responses and puttting them on post-it notes to add to the class chart. They are to start with 3's and continue through groups of 12.

4. Come back together as a whole class and discuss the ideas presented on the post-its. Remove any duplicates and complete the class chart. Talk about categories that were difficult to fill. Students can then fill out their individual charts for their Math Journal.

**Day Two:**

Review the chart and add any new ideas. Pose a problem with an example from the chart using the 2's. For example, "If I had 6 children, how many eyes would there be?" Draw a picture to show the answer. Write the appropriate number sentence: 6 x 2 = 12. Do a few more examples. Using interlocking cubes, show how to "discover" the 2's facts.

Continue until 12 x 2 = 24.

Have groups of students continue to work with the cubes and record their multiplication sentences on paper. One group can do 3's, another do 4's and so on.

<span style="font-family: 'Arial','sans-serif'; font-size: 10pt;">Gather students together with their charts and discuss the findings. Discuss the patterns demonstrated in each number fact. Review concept of multiplication as sets of groups (4 groups of 2 is the same as 4 x 2 = 8). Assign each group of stduents a multiplication sentence and have them create it with interlocking cubes. Then have them show what multiplication fact will come next. Discuss the pattern.

<span style="font-family: 'Arial','sans-serif'; font-size: 10pt;">**Day Three:**

<span style="font-family: 'Arial','sans-serif'; font-size: 10pt;">Review the multiplication sentences and the patterns from yesterday. Tell the students that they will be writing stories using ideas from their list of multiplication sentences. Review the class chart.

<span style="font-family: 'Arial','sans-serif'; font-size: 10pt;">On chart paper, show the students how to create a short story that coordinates with one of the multiplication facts from yesterday. Example: Once there was a turtle with 10 children. She wanted to buy them all shoes but she didn't know how many shoes to buy. Can you help Mrs. Turtle?

<span style="font-family: 'Arial','sans-serif'; font-size: 10pt;">Tell the students that, in pairs, they will be making pages for a multiplication class book. The front of each page will have a story problem and an illustration. The back of each page will have a multiplication fact that goes with the story problem.

<span style="font-family: 'Arial','sans-serif'; font-size: 10pt;">Assign a multiplication fact or facts to each pair of students and hand out paper. Allow time for students to complete their pages. Teacher editing may be necessary. Have students who finish early work on a cover. Assemble book for class.

<span style="font-family: 'Arial','sans-serif'; font-size: 10pt;">**Day Four:**

Because children learn in different ways, its helpful to include a diversity of approaches in lessons. The following game suggests a way to introduce children to multiplication as combining equal groups, often referred to as repeated addition, with the additional benefit of providing a visual representation of multiplication. In this game, the children interact with the idea of multiplication pictorially (by drawing circles and stars, symbolically (by writing the multiplication equations), and verbally (by reading the equations).


 * Circles and Stars:**

Collect the materials needed for the game: one die for each pair of children, a sheet of unlined paper for each child, a pair or scissors for each child, and two or three staplers.

Tell the children that you're going to teach them how to play a game called //Circles and Stars// that they will then play in pairs. To model how to play, invite a child to be your partner and join you at the board.

Begin by rolling a die and reporting to the class the number that comes up. Draw that many circles on the board, pointing out that you are drawing the circles large enough to be able to draw stars inside. Then have the child who is your partner roll the die and draw the appropriate amount of circles. Roll the die again, report the numbers to the class, and draw that many stars in each of your circles. have your partner roll the die and draw the correct number of stars. Ask the class to figure out how many stars each of you drew. Write the correct number of stars underneath each drawing. Play another round to be sure the children understand, and then ask the volunteer to be seated.

Tell the class that a complete game takes seven rounds. Demonstrate how to fold a piece of paper into fourths, cut it apart, and staple it into a booklet. Write //Circles and Stars// and your name on the front. Have the children count as you show that seven pages remain for playing. Tell the children that although they can see who wins each round by comparing how many stars each player drew, the winner of the game is the person who has more stars altogether in the booklet. After seven rounds, children are to figure out the total and record it on the cover.

Have the children make their booklets and play the game. Those who finish quickly can make a second booklet and play again.

When all of the children have played at least one complete game, show them how to record a multiplication equation on each page. First draw on the board a sample of three circles with two stars in each and underneath write //3 x 2//. Explain to the children that this is a way to write three sets of two or three groups of two with math symbols. Tell them that you can also read it as "three times two" and it means the same thing. Show how adding //= 6// tells how many stars there are in all. Write on the board the three ways to read 3 x 2 = 6:



Then ask the children to work with their partners and agree on the math equation to write on each page of their booklets. Tell them that as they write the math equations, they should read them out loud to each other in two different ways.

After all groups have played at least one game, have them work in small groups and compare the number of stars on each page of their booklets. Ask them to look for the different numbers of stars that came up, which numbers of stars on a page came up frequently, and the different ways they got them. Have children present their findings in a class discussion.



I was unable to put this into a power point but it can still be used with the document camera. It will walk you through showing your students how multiplication is related to addition (repeated addition). This is a concept that is easy for kids to understand.


 * Day Five:**

Investigating rectangular arrays introduces children to a geometric model for multiplication. The students investigate rectangular arrays as they research how to package candy. The "candies" are 1-inch square tiles that are packed one layer deep in rectangular boxes. Children use the tiles to identify various dimensions of boxes for different numbers of candies.


 * Candy Boxes:**

Collect the materials you need for this activity: color tiles (at least 12 per child), half-inch grid paper, scissors, and glue.

Distribute tiles so that each pair of children has twenty-four. Explain that the tiles are pretend candies that come in rectangular or square boxes and are always packed in just one layer. Tell the children that a sampler box has four candies in it. Ask each child to take four tiles and see how the candies might be arranged to fit into a box.

Typically in a class, students produce the two possible options. Draw them on the board. If only one is found, however, show the other and have the children build it with tiles. If children suggest an L-shaped arrangement, remind them that the box must be rectangular or square. This is a good opportunity to point out that squares are really special rectangles because each of their sides are the same length.

Write the dimensions //2 x 2// and //1 x 4// inside each rectangle you drew.



Read them aloud as "two rows of two" and "two by two" and "one row of four" and "one by four" and explain to the children that the numbers tell how many tiles their are on adjoining sides. Then, using half-inch grid paper, cut out the two different arrays and record the dimensions on each. These can be glued into their Math Journals.

Next, pose the research problem: Each pair of students is the design reasearch team of the candy company. The president of the company has asked for a report about the different boxes possible for six, twelve, and twenty-four candies. Give the children three directions.


 * 1) For each number, use the tiles to find the possible rectangles.
 * 2) Cut out each rectangular shape as you find it, using half inch grid paper.
 * 3) Write a memo to the rpesident explaining what you've learned about boxes for each quantity and what shape you recommend. include your cutout boxes with your memo.

Have children report their findings and recommendations in a class discussion.




 * Day Six, Seven and Eight:**

Independent Activities (Menu)

The independent activities provide children with a variety of experiences with multiplication.


 * Patterns in Multiples:**

You need: "Things That Come in Groups Charts" (made earlier with the class), 0 - 99 charts


 * 1) Choose an item from one of the "Things That Come in Groups" charts. List at least 12 multiples and write the multiplication sentences.
 * 2) Color in the multiples on a 0 - 99 chart. Then continue the pattern to the end of the chart.
 * 3) Write about the aptterns you see in the numbers on your list and on the 0 - 99 chart.
 * 4) Repeat for an item from a different list.


 * Candy Box Research (Version 1):**

You need: partner, 36 color tiles, half inch grid paper, bag with numbers (1 - 36)


 * 1) Pick a number from the bag.
 * 2) Using color tiles, build all of the rectangular boxes possible for that number of candies. Cut out each from the grid paper and label its dimensions.


 * Candy Box Research (Version 2):**

You need: partner or small group, 36 color tiles, half-inch grid paper, Math Journal
 * 1) Use color tiles to build the rectangular boxes possible for all the numbers from one to thirty-six. Cut each from the grid paper and label its dimensions. Glue into Math Journal.
 * 2) When you've completed your journal, write answers to the following questions:
 * For which numbers are there rectangles that have sides with 2 squares on them? Write the numbers from smallest to largest.
 * For which numbers are there rectangles that have sides with 3 squares on them? Write the numbers from smallest to largest.
 * For which numbers are there rectangles that have sides with 4 squares on them? Write the numbers from smallest to largest.
 * For which numbers are there rectangles that have sides with 5 squares on them? Write the numbers from smallest to largest.
 * Which numbers have rectangles that are squares?
 * How many squares are in the next larger square you can make?
 * What is the smallest number with exactly 2 different rectangles? Three different rectangles? Four?
 * Which numbers have only one rectangle? List them from smallest to largest.


 * More Circles and Stars:**

You need: a partner, 1 - 6 die


 * 1) Make a booklet and play Circles and Stars as you did before.
 * 2) Make tally marks on a class chart to show the number of stars you drew on each round. (1 - 36)


 * Billy Wins a Shopping Spree:**

You need: a partner, copy of story problem


 * 1) Make a receipt that shows how many items of each price Billy bought. Show the total spent and the credit he has left, if any.
 * 2) Extension - Find the combinations of differently proced objects that Billy could buy and spend $25.00 exactly.


 * Multiplication Stories:**

Write a multiplication story that follows two rules:
 * 1) It must end in a questions.
 * 2) The question must be one that's possible to answer by multiplying.

Solve your story problem as many ways as you can. Exchange papers and solve each other's problems.


 * How Long? How High?:**

You need: a partner, a piece of graph paper, a die


 * 1) Each player uses their own piece of graph paper. On your turn, roll the die twice. The first roll tells you how long your rectangular box (array) will be. The second roll tells you how high your rectangle (array) will be.
 * 2) Draw your array on your graph paper and write the appropriate multiplication sentence inside.
 * 3) Take turns until one of you can't place your rectangle because there's no room on the grid. Then figure out how many of your squares are covered and how many are uncovered. The winner is the partner with the most squares covered.

This is a really cool game but I was unable to show an example. Please let me know if you would like me to explain (Sandy).


 * Powerpoints:**

These are student made word problems. This could be a project you could do with your class if you had the time or wanted to get parents involved. The kids would just need to bring in their own props.




 * Links to great websites (I checked them out!):**

[|www.multiplication.com]

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 * Multiplication/Division Menu (TAG):**

If you need something for your higher kids, try this.



<span style="font-family: 'Arial','sans-serif'; font-size: 10pt;">
 * Activities that go with picture books:**

<span style="font-family: 'Arial','sans-serif'; font-size: 10pt;">**Envision Resources:** <span style="font-family: 'Arial','sans-serif'; font-size: 10pt;">

CGI Problems: