**This practical lab is designed to show students how they are already using math in their everyday lives and then help them improve those skills to make better decisions, improve problem-solving skills, and promote confidence. This lab is a learning community where participants share strategies, justify their reasoning, and interact with each other’s ideas.**

# MATH LAB NOTES

4/11/2023

Using Benchmarks: Lesson 8, Making sensible rules for adding and subtracting

Materials needed:

Decks of cards, 5 copies of p 93, 125, separate pages for our 4 math rules (properties), mancala boards (if you want to play at the end)

Warm up/game

Close call: Throw out 10s and face cards. Write “Ace = 1 and Joker = 0” on the board. Split the deck between 2 players. Each player picks 4 cards (randomly, off the top of a shuffled card pile) and create two 2-digit numbers that when added together come as close as possible to 100, without going over. Whoever gets closest to 100 for that round wins a point.

Collecting rules or “properties”

We will start to collect rules or “properties” that are always true in math. These should make sense. You don’t have to memorize rules you do not understand.

Write these questions on the white board. Let the class discuss each, come to a consensus, and give examples that support their claims. Give them time to come to the answer on their own.

Does order matter when adding numbers?

Does order matter when subtracting numbers?

Does order matter when multiplying numbers?

Does order matter when dividing numbers?

Post the 2 papers about addition and multiplication on the white board and ask students to write some of the examples they used to prove the property on each paper.

Adding: order doesn’t matter! (Commutative Property)

Multiplication: order doesn’t matter! (Commutative Property)

Now work on p 93 (in pairs). Again, let them try out examples so they can see if something is true or false.

Post the last 2 properties about multiplying by zero and multiplying or dividing by 1.

Ask students to write some of the examples they used to prove the property on those papers.

Zero: Anything multiplied by 0 = 0

Anything multiplied or divided by 1 = itself! (The value doesn’t change).

Practice p 125. Do #1 together. Print off, split up problems. Have each group explain 1-2 problems and tell us if they used one of these properties as a shortcut. Students who are comfortable can do all of them.

If extra time…

Mancala: (Find these in the big black filing cabinet)

https://www.youtube.com/watch?v=iSJk6CYsf6c

3/21/22

Today we generated, organized, and analyzed data in real world situations. We talked about how data can tell us a story. It’s possible to see the story if you can organize the data. Averages are one way we can make sense out of large amounts of data. We looked at an actual Provo City utility bill and tried to find the story the numbers told us. This is one of my favorite labs because we all receive and pay utility bills, but don’t always take the time to analyze them.

Averages; Data gathering, organizing, and analysis; My electric bill

Materials needed:

copies of electric bill worksheet (for each), calculators for all, 2 decks of cards, JCWs menu, measuring tape, laptop with videos ready, paper at each station to record data.

Warm up/game

None today

Averages

Data tell a story. To find the story, we need to gather, organize, and analyze data.

DATA GATHERING

Set up stations with a piece of paper at each to record the data. At each station you need to write your name and your data. Do each in pairs, if possible.

Height: find your height in inches

JCW’s menu

https://www.jcws.com/assets/jcwsmenu_apr2020.pdf (pictures, no prices)

https://www.allmenus.com/ut/provo/649601-jcws/menu/ (no pictures, but has prices)

Everyone place your order! Organize order so you can see exactly what everyone ordered and how much each item was. Calculate total for each person .

Card Sort! (Or some kind of physical challenge). Sort a pile of cards into little piles of 2s, 3s, 4, 5, J, Q, K, etc.). Time each person and record your times.

Leave a review of this video: Give it a starred rating, from 1 star to 5 stars. Have slips of paper that students can write on and then fold so they aren’t influenced by each other’s ratings.

Tongue singing choir

Elvis without music

DATA ORGANIZATION

Make a table on the whiteboard to record everyone’s names and the data for each station activity. Let students put the data in the table. (You’ll have to convert the times into decimals and can talk about how 30 seconds is like ½ a minute. You can either record it entirely in seconds or entirely in minutes).

Talk about averages, where we see them, and when they are helpful. (Average age, average income, average cost of living, average review). Talk through an example of height. What if we all stood on the shoulders of each other and measured how tall we are all together, then divided that into the # people in the tower. That’s an average.

Talk about doing that for each of the station activities. Assign different students to pick an activity and find total, then find average. Students fill in the table and check each other’s work for accuracy.

DATA ANALYSIS

Present to the class what you did and tell us a little bit about the story the numbers tell us. Who liked the videos? Who didn’t? Many of us are roughly the same height and one of us is a lot taller. How did that affect the average?

Amy’s electric bill

The city of Provo has gathered and organized data about my electric/utilities. Now it’s our job to analyze it and to see if there is a story to tell. What questions do you have?

What is a kWh? “Kilowatt hour”= a measure of electric energy equivalent to a power consumption of 1000 watts for 1 hour (10 100watt lightbulbs running for one hour).

Explain tiered pricing system (tiers 1-3). Why would the power company choose to price these tiers differently?

Provo city is different than a typical company that would you love you to buy more and more of their product (like a company that makes snack foods). As a city we need to cooperate to make sure there is enough water and electricity for everyone and that it’s priced fairly for different kinds of consumers. We want prices to encourage conservation and discourage waste.

What months did I use the most electricity? Why do you think?

What months did I use the least? Why? Does this give you any clues about the kinds of things in a house that use the most electricity? What other questions do we have?

Compare electricity rates by state

What appliances use the most electricity in a typical household?

Look at calculations of averages on electric bill.

3/14/2023

Today was pi day! Thanks to Shauna, we shared a delicious berry pie. We were talking about how pi is a number related to the relationship of a circle’s diameter to its circumference. When I drew a circle and asked the class if they knew the name for that distance, a student spoke up to say that it was called circumference and that he had learned about it by watching the movie, “Hidden Figures”, about a group of female African American mathematicians who worked with NASA at the height of the Space Race in the 1960s. That was a perfect shout out to an amazing woman (Katherine Goble Johnson) for Women’s History month!

Pi Day!!; Using Benchmarks: Fractions and Operations: Lesson 2 Half of a half (¼)

materials

Pie for all? String, scissors, different circles.

Graph paper, copy pp 31-32, print “challenge pages”

Discussion:

Pi day is March 14th!

What is pi? A ratio. The relationship between the circumference/diameter of a circle.

Helpful terms: diameter, radius, circumference

Use string to find the circumference of any circle, cut string. Then cut sections of that into diameter pieces. You should be left over with 3 diameter lengths and a bit left. Video. Tortilla, orange slice, clock, cookie, bowl/cup, lollipop, bagels, paper plate,

C=2(pi)r; Diameter=2r The only difference is the ratio pi!

game

Fraction war: each player flips over 2 cards and makes a fraction with the larger # as the denominator. Who has the larger fraction? Use ½ and ¼ as benchmark fractions and decide if your fraction is larger or smaller than ½.. If it’s too close to tell, then use a calculator to divide and get an exact decimal. Fraction War

(Remove J, Q, K. Game is easier if you also remove 7 and 9. A=1)

Discussion: (Teacher’s book p 26)

Yesterday I was waiting in line at the post office, where one fourth of the people had packages to mail. There were 8 people in line. How many had packages? (Use snap cubes or draw pictures). Eventually have a discussion that includes finding the whole, breaking it into 4 equal parts, finding the value of one of the parts. Also how many people didn’t have packages to mail? What fraction does that represent?

-on whiteboard, draw a rectangle and 12 stars. Who can show me ¼ or 25% of these?

Fractions, percentages, decimals

Look at student book, p 30. Each student uses graph paper to draw their own 10×10 grid. Shade in ½. Look at this grid below. Explain how you know that the shaded portion is one half, ½, 50%, or 0.50.. Now make another 10×10. Shade in ¼. Do the same. Try to make a connection between fraction and percent. It’s ¼ if you visualize 1 group out of 4 groups. It’s 25/100 if you visualize 25 groups out of 100 groups. How else could you break this down into other fractions?

Student book p 31-32

Do these with a volunteer. Group discussion: what were the steps?

03/07/2023

Today we talked about benchmark fractions, especially 1/2. We talked about may applications, including what it means to earn “time and a half” or how to know how much of your grocery budget you should have left if it’s only half way through the month. These seem like important concepts for career readiness.

2/8/2022

ONE HALF: Fractions Using Benchmarks; Fractions and Operations

materials

copy p8 for all (with Amy’s modifications), station papers taped around room, play pizza, tape measure, blank calendar, book with bookmark, sample money, fraction tiles, playing cards, post it notes.

Show me a half

Everyone think of a story or situation where you might need to find out what “half” is. Your story has to include the whole and the part. Example: I like gardening and will spend $50 on plants this season. I want to spend half on perennials and half on annuals. So I will spend $25 on each. Whole: $50, Part: $25. Fraction spent on annuals: 25/50 or “25 out of 50”.

Examples of things we might count or measure:

Buying things

Measuring cuts for a bookshelf or bench

Recipe for 8, but you only need to feed 4

My gas tank holds 20 gallons. I want to refill it when it is half empty.

Christmas: I have this much to spend on 2 nieces. How much will go to each?

What about “time and a half” for overtime?

Game

Benchmark fractions. Deck of cards, remove J, Q, K. A=1. Can play alone or cooperatively in a pair. It works best to be sitting next to each other. Choose a simple benchmark fraction, like ½. After shuffling the deck, put 6 cards face up on the table. Divide the rest of the deck evenly between the 2 players. Each player’s cards should be in a pile, face down, by them. Player 1 flips over the top card in their pile and sees if they can make an equivalent fraction to ½. Example: they flip over a 2 and they see there is already a 4 face up on the table. 2/4 is the same as ½, so they make that fraction by placing the 2 above the 4. Now player 2 flips over their first card and sees if they can make ½. Once a ½ fraction has been made, then those cards can be cleared from the table. You can also add cards together. Ex: 6/(10+2) is 6/12, or ½. So basically, once you flip over 1 card, you can either leave it on the table without making a fraction, or you can make a fraction with your card and others already on the table.

½ is a benchmark fraction

P8 student book (copy for all)

Station 1: It’s March 16th. Are we half way through the month yet?

Station 2: Measure the height of the door & the distance from the floor to the doorknob. Is doorknob half way from the floor to the top of the door? More than half? Less than half?

Station 3: See the book on table? I have read up to where the bookmark is. Have I read half? Less than half? More than half?

Station 4: I ordered a pizza all for myself and ate 5 pieces.

Station 5: Money: On March 15th, we will be half way through the month. My grocery budget is $500 for the month and I have already spent $230. Have I spent more than/less than half of my budget?

Challenge question. Do this on own, have volunteers help.

You get paid “time and a half” if you work on a holiday. If your normal hourly wage is $16, then what is time & a half? What if you get paid $17/hr? How much would you be paid for 8 hours of work?

Which is larger? P21

*Challenge: p22 half a million? Or test practice (p24)

Introduce fraction tiles. Come up with several fractions that are equivalent to ½.

Ex: ½, 2/4, 3/6, 4/8, etc. Encourage them to see a pattern and come up with a rule.

Numerator = Denominator divided by 2, or Denominator= Numerator times 2.

Draw big # line on white board. Zoom in between 0 to 1. If we kept zooming and chopping and zooming and chopping forever, we could have an infinitely smaller and smaller pieces. Crazy, right?

Write these fractions on post it notes: ⅖, ⅝, ¾, ⅓, 8/16, 6/12, 4/7, 4/9.

Pairs: Give each pair a fraction (one at a time) and ask them to to figure out if it is >,<, or = to ½ and then place it on the whiteboard # line.

Test Practice

For those who are ready & need more.

2/21/23, 2/28/23

Everyday Number Sense: Lesson 4, Traveling in Time; Lesson5: Meanings & Methods for Subtraction (part of lesson)

Game/warm up

A number line can show the difference between 2 numbers.

Counting up to/Counting Down from 100 with a number line:

Draw # line from 0-100, marking every 10.

Starting # (make a box for this)

Next to that box, draw 5 boxes that will contain a series of 5 numbers telling us to go + or – by multiples of 5. For example: +10, -5, +10, +20, -10,

Draw another box for the End #.

Students leave 1 of the boxes blank and the challenge is to figure out the missing #.

Have 4 of these on the board (written by students), then people can work them out on their own paper and we can review answers.

Variation: tell us to start at 100.

Materials:

Analog clock, p84, one for each student.

P64 Activity 1: Bday #s

I’ll write number line on white board, but they should also write number line on their paper.

Clocks & Time

Describe to your neighbor how you read time on an analog clock. That’s one way to do it. It’s essentially 2 number lines wrapped around a circle. Can you think of another way to show the passage of time? No right answer as long as it works.! Think creatively and present your idea to the class. Tell us why it might work well or not work well.

Another TIME challenge

Time Zones. What are they?

What time will it be in London if it’s 8am in Utah? What info do we need? Use internet to find time difference? 8am Utah time + 7 hours= 3pm London time

What if I’m in London and just had lunch and feel like calling home to Utah? If it’s 1pm in London, what time will it be in Utah?

When do we know that a problem is asking us to subtract?

Lesson 5 Activity #1 p 84. Let them talk about the differences between the 3 problems.

Take away (easiest to spot this one.) Use regular subtraction

Comparison (how much more…) Use bar model

Missing part (part-part-whole, use a bar model)

Use a bar model or a number line to show what the problem is asking for.

Learn how to use a bar model.

YouTube.com/watch?v=E3Z4WikFAf4

Do one of each of these kinds of problems together

Comparison: You usually cook for your family of 6, but tonight there are 9. How many more guests are coming?

Missing part: *Daniel and Peter have 450 marbles in all. Daniel has 248 marbles. How many does Peter have? (Part, part, whole) Get out manipulatives if you need to help figure it out.

Comparison: Who has more marbles? How many more does he have?

Comparison: Mary had 120 more beads than Jill, who had 68 beads. How many beads did Mary have? Then addition: How many do they have altogether? (Bar models)

Take away: I had 5 apples in the fridge and my kids ate 3 of them. How many are left?

Everyone make up one of each of these kinds of problems. We’ll try to solve it and show how we did it with a bar model or a number line.

If extra time, this game could be fun.

Cross Out Singles: Watch video. Multiple players, as many as you want. Some addition strategy and multiplication skills. Play 5 rounds, add up points to see who wins. Can draw their own board.

February 14, 2023

Everyday number sense, lesson 3, lesson 4

Goals:

Review place value in a way that you can pull apart a # (532 is 500 + 30 + 2) and then put it back together again.

Review how to say big numbers

More practice with word problems: deciding on a strategy, proving your reasoning

Materials needed:

100 grid for Euclid’s game, yellow manipulatives (rods & cubes), print pp 66-71 (1 copy), p 61 (4 copies?)

Game:

Euclid’s Game: Watch video. Play in pairs. Practice subtracting numbers until you use up all the numbers on the board. https://www.youtube.com/watch?v=CuocfDYpNoM

Show numbers on a number line and talk about how subtracting is just measuring the distance between 2 numbers. You can find an easy number between the 2 numbers and then measure the distances to that mid point, then add distances to find total distance. Ex: 54-37. I’ll pick 40 as my in between number. Distance from 37 to 40 is 3. Distance from 54 to 40 is 14. 3+14 = 17. So 54-37=17.

Birth year: understanding place value and distances in time.

Pick an important year to you: (birth year, or impt event) (4 digit number)

Write your number at the top of the page, Then divide rest of page into 3 sections.

1: Use manipulatives to build your number. Then draw it on your paper.

2: Break it into its different place value parts. 1952 is 1000 + 900 + 50 + 2

3: How many years ago was it? Can you show us how you know?

Review:

Place Value: how to read really big numbers (good video explaining) https://www.youtube.com/watch?v=DZ7fxH0C54Y

Write a 6 or 7 digit number and circle one digit. Students learn how to write what that digit represents. Example: 1,256,745. Circle the 2. Student learns to know that the 2 represents 200,000. Repeat for other digits within the large number.

Activity 2: How Long Ago? (p 66-71)

Take printed pp 66-71, cut each event into its own strip. Distribute 1-2 to each student. Follow directions on p 66 (fill in missing numbers, label the year of the event you chose and the current year, find the difference between those years, write an equation that shows how to find that difference.

Once everyone has completed those, then come together and have people share their event, place it on a class time-line and explain what they learned.

February 7, 2023

Today we got more comfortable with saying large numbers properly. I think being able to read, write, and say large numbers properly helps to understand and use those big numbers in work and life.

We also practiced kinds of mental math and allowed the students to find their own methods of figuring out estimates and exact amounts. See notes.

We had a nice discussion about how to spot and avoid being the victim of scams sent through email and text.

2/7/23

Everyday number sense, lesson 2

Goals:

Understand place value in a way that you can pull apart a # (532 is 500 + 30 + 2) and then put it back together again.

Learn how to say big numbers

Estimate how much a purchase will cost by rounding and then adjust to find the actual cost.

Materials needed:

Deck of cards, play money

Game

3-digit war: Flip over 3 cards and see who can make the larger 3-digit #. They win their opponent’s cards. (Remove 10s, face cards, aces, jokers).

Place Value: how to read really big numbers (good video explaining) https://www.youtube.com/watch?v=DZ7fxH0C54Y

Now try the game, but do 5 or 6 digit numbers and practice saying the number out loud.

Activity 1: Math in Line (p 22-23) Estimate how much your purchase will cost, then do mental math to adjust your estimate to the exact cost.

Work alone or in pairs.

Let them figure it out, then talk about the steps.

Round up or down?

By how much?

What’s your estimated total?

Is that higher or lower than the actual amount?

By how much?

Take estimated total and add or subtract by the amount you rounded.

Activity 2 (p 25) has them fill out a table to see how to round & adjust. Maybe we could come up with our own table together instead of using the book’s example?

How to detect a text/email scam

Why so many? Why not in paper mail as much? (cost)

What are the signs of a scam?

-Greatest desire/greatest fear

-Plays on relationships and money

-ACT NOW! Emergency!

-Click on a link. What is malware?

Good rules:

-Do I know the sender?

-Never click on a link, just go to the trusted website yourself.

-WAIT. It isn’t really an emergency.

-Never give your personal information.

January 31, 2023

Multiplying by 10 and 100 in an instant and knowing how to quickly find 2 numbers that add to 100 can help with all kinds of money, job, and life applications. These skills are important foundations of “math sense”.

1/31/2023

Everyday Number Sense, Lesson 2 (Mental Math in the checkout line)

Goals:

Identify 2 digit numbers that add up to 100 and find a pattern.

Find a rule for multiplying by 10, multiplying by 100

Materials:

Deck of cards, print multiplication grid, bring some rod/cube manipulatives, paper, pencils, calculators? Print p 22, 23

Game

(We’ll use this when we practice finding 2 digit numbers that add up to 100. One’s place should add to 10

Ten’s place should add to 9.

Number bonds to 10: Pair up with a partner. (Ace = 1, J=11, Q=12, K=13). Goal is to find 2 #s that add up to 10 or 20. Amy draw some number bond that =10, some that = 20. When I say GO, player #1 makes 3×3 array and player 2 starts finding pairs. Player 1 replenishes the array with new cards once matches are found and removed from the existing array. Player 1 and player 2 switch jobs. If you come to a point in the game where all 9 cards are laid out and there are no combinations that equal 10 or 20, then shift to where you can pick up more than 2 cards. Ex: King + 4 + 3

Number bonds to 9: same game, but only use Ace and 2-8.Do this? Or not.

Making rules, seeing patterns: How to quickly multiply by 10, by 100

Tools available: multiplication grid, manipulatives, calculators, paper

*Explain notation for multiplication: can use “x”, but can also use ( ). An “x” can be confused with a “+”, and we often use variables like x in algebra, so it’s good to be familiar with the parentheses.

What is 15 x 200 or 15(200) You may not be able to do that in your head yet, but you will soon!

Create a rule for multiplying by 10. Any number multiplied by 10 should follow this rule. Prove it. (With a table, with manipulatives, pictures). Let them get started themselves.

Create a rule for multiplying by 100. Any number multiplied by 100 should follow this rule. Prove it.

Does your pattern work for 15×10? How can we check your guess? How about 23×10?

Practice your rule on the following and then check your work:

12 x 100 10(45) 20 x 3 11 x 50

20(30) 20 x 23 10(45) 15(200)

Make a rule for how to find 2 numbers that add up to 100.

What helped you when finding a rule for how to multiply by 10?

Hear some rules, test them out, see what class thinks.

January 24, 2023

We had a great discussion about what mental math is for and practiced rounding to the nearest 10. We played an online game to practice this skill. https://www.mathnook.com/math2/math-lines-rounding.html

We practiced using rounding to figure out some real life word problems:

A friend and I go out for a night on the town. We see a movie and have dinner. The movie costs $9.50 per ticket. Dinner for the two of us is $43.50. About how much will our evening out cost each of us?

Sharonda is setting tables for a wedding. There are 19 round tables, each seating 8 people. She places 2 forks at each setting, so that’s about ___ forks in all. Estimate! Show how you arrived at your estimate. Can you tell if your estimate is bigger or smaller than the actual amount? Explain?

January 17, 2023

Everyday Number Sense, Lesson 1

Close Enough with Mental Math

Goal:

Use mental math strategies to estimate totals (rounding, making friendly numbers) #CCR

“friendly numbers” are easy to work with (5, 10, 15, 20, 25, 30, … 100)

Get there by rounding, get there by adding #combinations (4+6, 7+3)

Understand commutative property of addition

Materials:

Deck of cards for each pair, print p 12, 18 for each student,

Game (Rounding)

Online game: Math Lines Rounding: shooting ball with a number on it to another ball with a multiple of 10.

Teach rounding (show bike going up/down hill).

Round to the nearest WHAT? If 10, then think of multiples of 10 (10, 20, 30, 40…) That’s your beginning and end point. That’s the only part of your original # that matters. If you have 2,345 and have to round to the nearest 10, then only focus on the numbers in the 10s place and lower. In this case: 45. Beginning point =40, End point is 50. Is 45 on the up slope? Is it at the midpoint? Is it on the downslope? 45 is at the top of the hill (midpoint) so it gets to ride down the hill to 50.

Opening discussion (Teacher’s book, p 10)

A friend and I go out for a night on the town. We see a movie and have dinner. The movie costs $9.50 per ticket. Dinner for the two of us is $43.50. About how much will our evening out cost each of us?

Figure this out without pencil and paper. Make estimates in your heads and be ready to tell us how you figured it out. Instead of getting exact answers, today we will make estimates that are “close enough”.

-Students share methods.

-Point out rounding up or rounding down, explain if needed.

Activity #1 How much?

In notes, find blackline master 4 with problems. Student options are on p 6. Instead of A, B, or C have them raise a colored piece of paper. a=yellow, b=blue, c=pink. Observe who is getting the answer easily and who needs help. (use zoom chat for online group)

Practice: It’s About… (p 12)

Sharonda is setting tables for a wedding. There are 19 round tables, each seating 8 people. She places 2 forks at each setting, so that’s about ___ forks in all. Estimate! Show how you arrived at your estimate. Can you tell if your estimate is bigger or smaller than the actual amount? Explain?

Which students know how to quickly multiply 8×20?

Extra practice:

Could do p 18 (finding combinations of $ that equal 10 )

January 10, 2023

Today we focused on mental math: rounding and being able to add “friendly numbers” in our heads or with minimal paper & pencil work. We talked about how this can be useful on the job or in everyday life. Sometimes it’s nice to add something in your mind before calculating something exactly to know if your answer is “in the ballpark”. I’ve caught my own mistakes many times because I knew approximately what the answer should be. We played a card game that asked us to find pairs of cards that added up to 10 or 20 and had a great discussion about why our pairs worked out perfectly when our only cards were 1-9, but how it didn’t work out perfectly when we added in 10, 11, 12, and 13.

November 15, 2022

#math. 11/15/22

Keeping Things in Proportion

Lesson 8: Playing with the Numbers

Materials needed

Student books, calculators, cubes or other small manipulatives (will need these to show fact families and to cover bingo card), paper for bingo sheets, print p 106

Multiplication fact families

Division is just the inverse of multiplication. It undoes it. If multiplication is like tying a shoelace, then division is like untying the shoelace. Show this with cubes.

Roll a dice twice. (4 and 6) Find multiplication family. Ex: 4×6 = 24

Write down 4 mult/div facts about those 3 numbers.

Let’s get a total of 12 families on our board.

Now create a BINGO sheet (5×5 grid). Put one mult/div fact in each square.

I will call out a mult/div fact. If I call yours exactly, you can cover it, but you can also cover it if it’s in the same fact family. Ex: If I call out 24 divided by 6 = 4, then you can also cover 4×6=24

Ratios show a relationship

What are a few examples?

Get on computer to find more relationships with currency or changing from metric system to “imperial system”.

1 foot = 12 inches

5280 feet= 1 mile

4 Tablespoons = ¼ cup

$4 = 1 gallon

$1 = 0.97 Euro; $1= 0.85 Pound sterling

We can call these relationships “proportions”. We’ve talked about them in recipes, mixing OJ concentrate. (It’s ok to double the salt if you’re doubling everything else. If you only double the salt, but not everything else, then that relationship is off and it will not taste right).

How to tell if two ratios are equal

Activity #1 (p98) Cross product property. Multiply top # of the first by the bottom # of the second. Then multiply to bottom # of the first by the top # of the second. If the products are equal, then the ratios are equal. (see p 98)

Let’s try this out. (see p 98)

Finding missing # is a proportional relationship

Activity 2 (p 99):

What value for the missing # would make the proportion true? Practice using cross product property as well as multiplication fact families (inverse relationship of mult/division).

Practice p 103, 106 (word problems)

In person lab: We didn’t get to p 106 (word problems), but we did everything else. The multiplication/division fact families bingo game was great. The concept was right within reach for everyone, but not so easy that it was boring. Finding the missing number in a proportional relationship was a pretty difficult concept for most, so it was really nice to have 2 volunteers there to help students individually.

November 8, 2022

We reviewed the ideas of dimension, perimeter, and area. We practiced situations where a length or width is unknown, but we have perimeter and/or area and found ourselves dipping our toes into algebraic concepts. During the last 30 minutes, we split into groups with graph paper and measuring tapes to draw, measure, and calculate the area of different tiled sections in the Provo library. One group measured the bathroom/drinking fountain area just outside the PR office, and another group measured the tile on the stairs leading up to the ballroom. I thought it was a good, practical, problem solving activity and I think the students enjoyed it.

November 1, 2022

Dimensions, Perimeter, Area

Build & draw a 2D rectangle

Find dimensions, perimeter, area

Combine all shapes to make a new one. Analyze.

Test practice

Materials needed

Yellow cubes, graph paper, tape/magnets to post shapes, copies of p 60 test practice, tape for composite shape, scissors (1-2)

Dimensions, perimeter, area (Use Google Slides)

x

x

Get out cubes, have each person build a rectangle out of 24 cubes.

Dimension: a measurable extent of some kind (length, width, height). Discussion of 2D and 3D? What does that mean? (Example: 3D movie, 3D printer)

Perimeter: measure of the distance around the 2D shape. Examples of usage:

To discourage cats from entering your property, sprinkle some cayenne pepper, ground coffee beans, dried mustard seeds or dried rosemary at the perimeter of your yard.

Shop the perimeter of the store — that’s where most of the unprocessed foods are placed.

Black-clad guards roamed the internal perimeter while others manned the walls of the compound.

Area: measuring the space inside the 2D shape.

Show dimensions, perimeter, & area of the shape I created.

Now ask students to sketch the shape they made on graph paper and tell us the dimensions, perimeter, and area. (Zoom students may need to sketch their own graph paper).

Create a chart on board comparing dimensions, perimeter, and area of different shapes. (Areas will all be the same).

Test Practice, p 60

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Work alone or with a partner.

Composite shapes

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Talk about taking all the rectangles students created and making a new, big shape. Use cubes to build robot superheros and watch retro clip of 1980s Voltron cartoon. (Just for fun!) Is it easy to predict the area? Is it easy to predict the perimeter? What can we do to break this large, complicated shape into easier chunks?

Test practice, p 69

Especially #5, #6.

VIRTUAL MATH LAB

I used sentence.yourdictionary.com to find sentences where the words dimension and perimeter were used online. I thought it was a nice way to show how mathematical concepts are used in everyday language. We made rectangles out of cubes and talked about dimensions, perimeter, and area, and practiced calculating them. We noticed that rectangles with the same # of cubes sometimes had different perimeters, but always the same area. We explored why that might be. At the end, we talked about composite shapes (several shapes come together to make an unruly/difficult to measure shape) and how you might break it into its smaller shapes to calculate the area. In my in-person lab, we made robot creatures out of our cubes and then watched a retro trailer for the 1980s cartoon, Voltron, about many “shapes” coming together to make a larger creature.

We introduced some geometry today. The part where we identified shapes and named them was instructive, but maybe a bit tedious. It got more fun when we created designs using geometry tiles and then tried to give instructions to a partner on how to draw it (without looking). Here are the notes!

Over, Around & Within: Geometry groundwork: shape names, vocabulary for describing shapes and angles

Materials needed

Dice, ice breaker game, copy of “Smart” poem, fake money, paper/pencil, collection of objects to sketch, copies of “shape description chart”, geometry tiles

Game

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D-Icebreakers D-ICE BREAKERS questions

Warm up

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Poem: “Smart” by Shel Silverstein: Where the sidewalk ends (get out fake money for this), print poem for all and have them show the money as the poem goes.

Activity #1: Making a Mind Map p 2)

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Make a mind map using words, numbers, pictures, or ideas that come to your mind when you think of “GEOMETRY” and “MEASUREMENT”. Write your own. Group discussion.

Goals: recognize and describe shapes, find perimeter of rectangles & volume of rectangular solids, make drawings to scale, use linear, square & cubic units, use spatial reasoning to solve problems, make generalizations about 2D and 3D shapes (p 8 student book)

-Start vocabulary list in your notebook (acute, obtuse, parallel, perpendicular, right angle, congruent,

Activity #4: Sketch and object and shapes within (p 3-4)

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Amy will have a collection of objects for in-person lab. Virtual students will find something in their home to sketch. Amy do this first, to demonstrate, then students will do. 1. Sketch (simply) your object, 2. What shapes make up your object? What can you find? 3. Name or describe shapes. How many sides? Angles different/same? Parallel/perpendicular.

Shape description chart (p 11-12)

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As you do activity #4, start to fill in “shape description chart”(see pdf in lesson plan supplements). Can I get this to my virtual students? Start to define angles (right, acute, obtuse), talk about equal side lengths, parallel, perpendicular. Sign for 90 degree angle, hash marks for equal sides on a shape or angle. “Congruent” means “equal” for sides or angles. Begin to mark up the shapes with with these descriptions. (Teacher’s book p 223 gives names of shapes. I needed help with the names of the different triangles!)

Activity 2 Sharing Secret Designs (p 13)

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Lab: use geometry tiles and work with a partner.

Virtual: Amy will make a shape with geometry tiles, cover it, and then give descriptions and have students try to draw it. Now try with a partner. One person create a design with 3-6 shapes. Then they give short statements to get the other person to be able to draw their shape. Use geometry terms (shape names, parallel, perpendicular)

Test practice, p 18

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Good for practice with vocabulary.

We didn’t complete the entire worksheet, but we were able to figure out what our budget to buy a car would be, and we calculated what the interest would be for car #1 after 4 years. The students naturally wanted to talk about needs vs wants and how to get by without having to take out loans.

Place value, converting percentages/decimals/fractions, finding discounts & applying tax

Materials needed: Dice, calculators

Game: Place Value

In this game, you roll 2 dice and create numbers that need to find a place on an ascending number path. The video explains it well. 13 squares. Lowest possible is 11, and highest is 66 for our six-sided dice. You could have individuals work alone, or have 2 people work together, taking turns rolling the dice and deciding where to place the number. https://www.youtube.com/watch?v=Bd6uTclUHr8

Variation: Roll 3 dice, make 3 digit numbers. Decimals allowed. 35.2, 15.3. Which is easier and why?

Make a table with 3 columns: Percentage, decimal, and fraction. Give one number and practicing changing from one form to another. Students will need to get comfortable using a calculator to change a fraction to a decimal.

Application:

Calculate how much you’d save , then subtotal, then add 7% tax, then grand total. Teach students how to find percentages with a calculator (560 x .35).

$560 with a 35% discount

$900 with a 12% discount

$680 with a 17% discount

Delmy and I had a wonderful one on one tutoring session with lots of practice for adding fractions with unlike denominators.

Example problems:

With all adding and subtracting of fractions, the goal is to get each fraction to have a common denominator. Sometimes you only have to change one of the terms. Other times you have to change both of the terms.

Examples of changing one of the terms:

1/5 + 2/15 (can change 1/5 to 3/15)

7/8 + 1/2 (can change 1/2 to 4/8)

Examples where you have to change both terms:

5/8 + 7/6

You could make the denominator 48 (8×6), but I think it’s a good practice for students to make a list of multiples and find the “least common multiple” to use as the denominator. Multiples of 8 are 8, 16, 24, 32, etc. Multiples of 6 are 6, 12, 18, 24, 30. You can choose 24 as the denominator (since they are multiples of both 6 and 8) and then it also makes it easy to see what you need to multiply 8 by to get 24 (it’s the 3rd one on your multiple list, so you multiply by 3).

Then, you can work on simplifying your answer! That requires a student to find the “greatest common factors” and to divide numerator and denominator until it can’t be further simplified.

There are so many steps with this kind of problem. It can be a tricky process for students and it’s great to see them stick with it!

½=2/4= 3/6= 4/8=5/10=6/12=0.50

¼ = 2/8=3/12=4/16=5/20=0.25

⅓ = 2/6 = 3/9=4/12=5/15=.033

1=2/2=3/3=4/4=8/8=10/10Make playing cards out of each fraction the students found. (Should be 5-6 of each fraction group.

Play “spoons”. 4 cards to each player. The goal is to hold only cards that are equivalent fractions and to grab a spoon at the end of round. The dealer takes a card off of the draw pile and, since they must only have 4 cards at a time, decides which card to “discard”/pass to the next player. The next player takes that card, decides which card to discard/pass on, etc. Instead of passing cards to the original dealer, the card goes in a “discard pile” at the end. Players can draw and pass cards as quickly as they want, so cards may build up between players. That’s ok, but players can only look at one card at a time from their building pile. The first person to have all 4 cards be equivalent fractions gets to grab one of the spoons on the table. As soon as one person grabs a spoon, then the rest of the players must grab a spoon as quickly as possible. There will be one less spoon than the number of players, so someone will always be left without a spoon. You can either have that person be “out” or can give them a certain amount of times to be left “spoonless” before getting out. Once everyone has grabbed spoons, you must check that the first person to grab a spoon actually did have 4 equivalent fractions. If they didn’t, then that person loses.

Keep the equivalent fractions posted on the whiteboard so students can refer to it when deciding whether to keep or discard a card during play. (We decided to color-code these equivalent fractions on the 3×5 cards to make it a bit easier as we played the game).Make chex mix for students.

“Chex mix tripled” lesson supplement (like student book, p 41). We gave the recipe to the students and asked them to triple it.

August 23, 2022

We used cubes, rods, and 100 squares (manipulatives found in the PR office) to solidify place value concepts and to prove if two equations are equal (see examples below)

Starting from a certain number, give step by step directions like “add 10, subtract 11, add 55. See if we come up with the same answer at the end. Use manipulatives to add and subtract. Practice having to “cash in” a 100 square for 10 rods, etc.

Equal or Not Equal

Give everyone at least 2 3×5 cards. They need to write one equation that is equal and one that is not equal. For example: 52+23=100-25 (true) and 21-12=3+11. (not true) We will shuffle up the cards and ask people to prove with manipulatives.

-When you have an equality laid out on the table, ask questions like, what happens when I add 5 to both sides? Is it still equal?

Imagine you are volunteering at a community event and you’ve been asked to stuff 1000 envelopes and peel 50 pounds of potatoes. How will we figure out before hand how long this will probably take? Let students reason it out & hopefully decide to time someone doing a smaller amount and then use that rate/ratio to predict the larger amount. (Student Book pp 25-27). Bring paper, envelopes, potatoes, and a peeler.

Application: Wasting water (Student book: Keeping things in Proportion, p 30).

Go to restrooms by PR office and take samples of dripping faucets. Let students decide how to set this up. How many gallons of water might we waste in an hour? A day? A 30 day month? A year?

Set up a conversion chart:

3 teaspoons = 1 Tablespoon 60 seconds = 1 minute

4 Tablespoons = ¼ cup 60 minutes = 1 hour

4 c water = 1 quart

4 quarts = 1 gallon

August 16, 2022

We reviewed place value and especially focused on the numbers to the right of the decimal point. We practiced interpreting 1/10, 1/100, and 1/1000 and learned how to put decimals into words. For example: 543.278 is five hundred forty three and two hundred seventy eight thousandths. We also practiced expanding out decimals into their components. For example: 543.278 = 500 + 40 + 3 + 2/10 + 7/100 + 8/1000. After this, we went through some test practice questions on p 85 of Using Benchmarks: Fractions and Operations (student book). We filled the rest of the class time working through these and teaching mini lessons where more support was needed. I encourage you to always use fraction tiles first to understand the problem more deeply, and then we learned how to do the problem without fraction tiles.

August 9, 2022

We talked about number lines and measuring distances on the map. Worked on figuring out the distance between two number/locations based on where they are on the line/map route.

August 2, 2022

Keeping Things in Proportion:

Materials needed

Deck of cards for everyone, Life cereal boxes.

Warm up/game

Pyramid: This link will explain it best. You form a pyramid with your cards and find pairs that add up to 10. There are specific rules involved that make it challenging and fun. Single player game, but you can compare scores and have a ranking on the whiteboard. One clarification: you can only use completely uncovered cards. One exception: if, by lifting one card, you can make a pair with the card in your hand and a card that is just now uncovered, then you can use that card that was partially covered before. (Remove J, Q, K, Jokers. Aces = 1)

Is buying in bulk a good deal?

Last week we imagined buying 2 pkgs of berries for $5 and then thought, what if I need 4 pkgs or 6 pkgs? The price for 2 pkgs never changed, but the amount that you decided to buy changed.

Now we are going to compare buying 2 pkgs of the same item, but at different prices. Sometimes you’ll see something labeled “family size” or “party size”, or will go to a bulk store like Costco or Sam’s club and see that you’re getting more, but you’re also paying more. How do you know if it’s a good deal?

Life cereal boxes. Find ratio ($/oz) for each box, then compare the fractions to see which is cheaper. Get fraction tiles to remind us that a larger denominator means a smaller fraction.

I have the Life cereal boxes for you here. Let’s go on Walmart.com to look up other prices.

Life Cereal:

Normal box: $4.59/18 oz

Costco box: $10.49/62 oz (take time to read the box to make sure they can find this info)

Doritos:

Normal bag: $3.48/9.25 oz

Party size: $4.98/14.5 oz

When is buying in bulk a good idea?

When is it NOT a good idea? (You need to know that you’ll like it and not waste it. Is it going to go bad if you don’t use it quickly enough and have to throw some away? Will buying more actually mean that you’ll just use more than you would have normally?

Test practice

Student book p 33. Give students time to try some of these.

July 26, 2022

Randy has had his right arm in a sling for the last while and I asked him if he is right or left-handed. It turned out that he is left handed (or ambidextrous) and that out of 7 people in the class today, we had 3 lefties! We had to make a fraction out of this information (3/7) and then had to search the internet to find out that, on average, 1 out of 10 Americans are left handed. When comparing 1/10 and 3/7, we had a nice discussion about fractions and how a large denominator means that then value of the fraction is pretty small. It feels like this is a concept that every one of our students needs frequent help with. Drawing pictures and using fraction tiles can help! The rest of the time, we used the grocery store ads from last week to practice making fractions and then finding equivalent fractions. Ex: 2 pks of berries for $5. How much would 6 pks cost? If we multiply 2 x3, then we get 6 pks and $5×3 = 15. The student book gave us a good amount of practice with problems like this. Toward the end, the class figured out how find a person’s heart beats per minute using a stopwatch and taking a smaller sample (15 seconds) and then multiplying up. I was proud of Delmy for being brave enough to come to the board and show her reasoning. Alma, Randy, and Moroni all volunteered answers during class. Jade asked a really good question about dividing $3 by 2. We got out the “play money” and demonstrated splitting one of the dollars into 4 quarters before trying to divide the whole thing into 2 parts. I love it when our students feel comfortable asking their questions in the moment they have them. That means that they resolve it then and there and don’t have to feel lost for the rest of the problem.

July 19, 2022

Lesson 1: A close look at supermarket ads (Keeping Things in Proportion)

Materials needed:

Grocery store ads with ratios, copies of entire lesson 1 student book for each student (pp 8-20), 2 dice with fractions on the sides for game, scissors, tape

Warm up/game

Fraction tiles: “COVER”

Fraction tiles: “UNCOVER”

Need a dice/cube with sides: 1/12, 1/12, ⅛, ⅛, ¼, 1/2

This lesson/chapter is so good. I think I will copy off the entire student section for each student so they can do the exercises and keep them for reference.

Comparing numbers

Teacher’s book p 21

Examining the relationship between #s is an important math skill.

Give me some true statements about the following number pairs: (100, 25) (500, 50) (2, 6) (3, 9), (4, 12) You can use +/- statements, but also multiplication and division statements. (Remember our mult/division math fact families?) We’re going to talk about ratios today: a way of comparing numbers that relies on mult/division comparisons.

Write vocab

Definitions: ratio: a way to compare two numbers or quantities.

For every “a” of one thing, there are “b” of another thing.

Ex: “1 teacher for every 25 students”, 60 miles per hour, $4/lb, $5 a gallon.

Proportion: a statement that two ratios are equal. Ex ⅓ = 2/6

Per: “for each” or “for every”

A bit different than part/whole

A fraction can be used to show a part/whole relationship, but it can also be used to show a proportion. This is a different emphasis than in our previous discussion of fractions. (Teacher book p 13). A relationship is preserved, even though numbers may change.

Activity 1

Supermarket ads, create the ratio (round up to nearest whole #: $3.79 becomes $4), then create equal ratios and show picture of the ratio (see example in teacher’s book, p 14). What would be a better deal? What would be a worse deal?

We pulled out fraction tiles for the first time today. The “cover” game wasn’t as fun or useful as the “uncover” game. I like that it invites students to change one fraction in for another (change ¼ into 3/12). Just handling the tiles and becoming fluent with changing things back and forth is a great step toward literacy with fractions. Fraction tiles are available in the Project Read office for any tutoring pairs to check out and use.

July 12, 2022

Gathering and classifying data (Book: Many points make a point, lesson 2). We gathered data about our students’ favorite foods and had them analyze the data and group them into categories like raw/cooked, plant/animal based, etc. The groups made posters with their information and then calculate the fraction or percentage for each category. We talked about why it’s hard to make categories that work and how to make them better. (Each data item needs to clearly fit into one and only one category). We talked about different situations where data is gathered and which parties might care about the information (advertisers, health providers, insurance companies). We ended with a discussion about converting fractions to percentages and some want to work on that more next time.

July 5, 2022

We introduced the idea of data and graphs today. (Math book titled Many Points Make a Point: Data & Graphs). The teacher’s book suggested creating a “mind map” where you brainstorm everything the students already know or can observe about an assortment of graphs. I wasn’t sure how this part would work, but it was actually awesome. Some responses included: time (change over time), heartbeat, numbers, news, and even a “scope” (comparing the appearance of a pie chart to the scope on a rifle). Next, we started gathering our own data sets and analyzing our results. We laid out 20 items of clothing on the tables and gave each student 10 post it notes. They selected 10 items of clothing and wrote down the country of origin on a post it for each shirt. They organized the post-its and were able to see which countries appeared most often. We labeled each country on a map of the world, made more observations, and talked about why we care where clothes are manufactured, economic and political implications, and why clothes are more likely to be made in some countries. Perry has grown in confidence and often has really innovative and intuitive ways to solve problems. When looking at his data, he was able to say, “Asia manufactured 80% of the clothing compared to 20% in Central America”. I praised him and challenged the other students to make sentences with numbers that were true about their data sets. We talked about how having a larger data set can increase confidence in the reliability of the statements we make. Using 10 shirts vs 100 shirts might solidify the trends we saw. Great lab today! Our textbooks give such great, hands-on lesson plans.